A spin-filter for polarized electron acceleration in

plasma wakefields

Yitong Wu1,2, Liangliang Ji1,3*, Xuesong Geng1,2, Johannes Thomas4, Markus Büscher5,6,

Alexander Pukhov4, Anna Hützen5,6, Lingang Zhang1, Baifei Shen1,3,7§, and Ruxin Li1,3,8†

1State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of

Sciences, Shanghai 201800, China

2Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049,

China

3CAS Center for Excellence in Ultra-intense Laser Science, Shanghai 201800, China

4Institut für Theoretische Physik I, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany

5Peter Grünberg Institut (PGI-6), Forschungszentrum Jülich, Wilhelm-Johnen-Str. 1, 52425 Jülich, Germany

6Institut für Laser- und Plasmaphysik, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany

7Shanghai Normal University, Shanghai 200234, China

8Shanghai Tech University, Shanghai 201210, China

Abstract

We propose a filter method to generate electron beams of high polarization from bubble and blow-out

wakefield accelerators. The mechanism is based on the idea to identify all electron-beam subsets with

low-polarization and to filter them out by an X-shaped slit placed right behind the plasma accelerator.

To find these subsets we investigate the dependence between the initial azimuthal angle and the spin of

single electrons during the trapping process. This dependence shows that transverse electron spins

preserve their orientation during injection if they are initially aligned parallel or anti-parallel to the local

magnetic field. We derive a precise correlation of the local beam polarization as a function of the

coordinate and the electron phase angle. Three-dimensional particle-in-cell simulations, incorporating

classical spin dynamics, show that the beam polarization can be increased from 35% to about 80% after

spin filtering. The injected flux is strongly restricted to preserve the beam polarization, e.g. <1kA in Ref.

[27]. This limitation is removed by employing the proposed filter mechanism. The robust of the method

is discussed that contains drive beam fluctuations, jitters, the thickness of the filter and initial temperature.

This idea marks an efficient and simple strategy to generate energetic polarized electron beams based on

wakefield acceleration.

I. INTRODUCTION

Wakefield acceleration driven by either laser pulses (LWFA) [1] or electron beams (PWFA) [2] is

advantageous due to its large acceleration gradient, reaching near GeV/cm [3] which is about over 3

orders of magnitudes higher than traditional accelerators [4]. Generation of GeV-level electron beams

from both LWFA [5-7] and PWFA [8-10] have been realized recently, among which 8 GeV electron

energy is obtained via LWFA [6] and several GeV via PWFA [9, 10]. These results pave a path to

facilitate table-top accelerators and secondary light sources. An ultimate goal of wakefield

acceleration is the pursuit of future electron-positron colliders [11,12]. Particularly, the electron and

positron beams are favorable to be spin-polarized for the colliders. There are at least three main

advantages for applying polarized beams: (i) The cross section is enhanced for certain reaction

processes [13-16]. (ii) Unwanted background processes and reaction channels can be suppressed with

appropriate combinations of the electron and positron beam polarizations [14,15]. (iii) By choosing

suitable observables in the scattering processes, additional information such as quantum numbers and

chiral couplings can be obtained [14,16]. Furthermore, energetic polarized electron beams are also

used to generate polarized photons [17] and positrons [18] as well as to study material science and

nuclear physics [19-21].

Conventionally, high energy polarized electron beams are obtained either from storage rings [19,20]

based on radiative polarization (the Sokolov-Ternov effect [22], similar effects using ultra intense

lasers are shown in Ref. [23]) or by extracting polarized electrons directly (via photocathodes [21,24],

spin filters [21,25] or beam splitters [21,26]) for subsequent acceleration in Linacs. Recently polarized

electron generation based on plasma wakefields has been explored in simulations [27-29]. The idea

includes preparation of pre-polarized targets via the photodissociation of hydrogen halides [30-34].

The pre-polarized electrons suffer from depolarization in plasma wakefield due to the spin precession

in self-generated magnetic field associated with the injected electrons. In order to preserve the electron

polarization during the entire acceleration process, strong limitations must be imposed on the beam

parameters, the plasma density as well as the injected beam flux. This can be mitigated by either

restricting the electron flux to <1kA at laser amplitudes a<1.1 as proposed in Ref. [27] or employing

vortex wakefield structures [28].

In this paper, we proposed a filter mechanism by selecting targeted electrons out of the whole

accelerated beam to achieve polarization>80%. The mechanism largely removes the flux limitation

issue on injected electrons. It relies on the fact that the beam polarization is dependent on the

spatial/momentum angles-a universal phenomenon in density down-ramp injection. Specifically, in

PIC simulations incorporating spin dynamics we show that spin precession in wakefild is significantly

suppressed when the initial spin is parallel or anti-parallel to the local azimuthal magnetic field,

leading to beam polarization dependence on the transverse coordinate angle in the plasma bubble. We

then propose to select electrons from the related region in the transverse phase space with an X-shaped

slit, which successfully maximizes the polarization of the accelerated beam. This unique feature has

not been revealed by any previous studies and applies to transverse polarization.

II. SIMULATION METHODS AND SETUPS

In order to demonstrate the spin-filter mechanism, we carry out full three-dimension (3D) particle-in-

cell (PIC) simulations with the code VLPL (Virtual Laser Plasma Lab) [35]. To investigate spin

dynamics, we integrate the spin precession into PIC code following the Thomas-Bargmann-Michel-

Telegdi (T-BMT) equation [36]:

��/�� = � × � (1a)

� =�

��

�

�� −

�

���×

�

�� + ��

�

��� −

�

����(� ⋅ �) − � ×

�

�� (1b)

Here e is the elementary charge, m the electron mass, v the electron velocity, and γ=1/(1-v2/c2)1/2 the

relativistic factor. s represents the normalized particle spin-direction vector with |s|=1 following

Ehrenfest’s theorem [19]. ae=(g-2)/2≈1.16×10-3 (gyromagnetic factor g), while the vector Ω is the

precession frequency. As mentioned in Refs. [27-29], other effects such as the Stern-Gerlach force (S-

G, gradient force), the Sokolov-Ternov effect (S-T, radiative polarization) [22] and electrostatic

Coulomb collisions [37-38] are negligibly small for wakefield acceleration. Comparing the S-G force

to the Lorentz force of wakefield acceleration |FSG/FL|~|∇(S·B)/γe2cBme|~ћ/λmecγe

2~10-7 [19,28]

suggests the drift motion caused by such a force is small enough to be neglected. With regard to the

S-T effect, the typical polarization time is about Tpol,S-T=8me5c8/5√3ћe5F3γe

2 [19,28]. For typical

wakefield acceleration (γe~103 and field strength F~1016V/m), one has Tpol,S-T~1μs, corresponding to

about 300m acceleration distances, which is much larger than typical wakefield acceleration distance.

In all the simulations, in order to ensure high injection efficiency [39-41], a transversely pre-

polarized target (along the +z axis) with a density bump is assumed. According to previous works [27-

29], such a plasma density is usually parametrized by a ramp-shaped profile, n(κ)={[α-Θ(κ)]Θ(L-

κ)cos(πκ/2L)+Θ(κ-L)}n0, where Θ(x) is the step function, κ=x-xp, xp=36μm, L=16 μm, n0=1018cm-3.

α=np/n0=4 is the ratio between the peak density of the ramp and the background density. The driver

beam (laser or particles), propagates into a moving window of 48μm(x)×48μm(y)×48μm(z) size and

1200×480×480 cell number with 6 macro-particles in each cell. The laser beam is linearly polarized

along the y-axis, following a Gaussian profile, EL=aw02/w2(x)sin2(πt/2τ0)sin(πr2/λR)exp[-r2/w2(x)],

with normalized peak amplitude a=eE/mωc, r2=y2+z2, wavelength λ=800nm, w(x)=w0{[(x-

xp)2+xR2]/xR

2}1/2, R=(x2+xR2)/x, Rayleigh length xR=πw0

2/λ, width w0=10λ(8μm) and duration

τ0=10λ/c=26.7fs, respectively. The electron driver beam also takes a Gaussian profile [8] nb(r,ξ=x-

ct)=nb0exp(-r2/2σr2-ξ2/2σl

2), where σl, σr are the longitudinal and transverse beam sizes and

nb0=1.5×1019cm-3 denotes the peak density of the driving beam, respectively. The simulation time step

is Δt=0.01λ/c ensuring that the largest precession angle in each time step |θs,max|=ΩmaxΔt~0.02πBmax (B

is normalized by mω/e) is sufficiently small (|θs,max|<<2π).

III. RESULTS

The electron density distributions of a typical LWFA and the corresponding transverse fields based on

density down ramp injection are displayed in Fig. 1(a). The results are collected for a=2.5, α=4 and

n0=1018cm-3. A sphere-like bubble is generated by the laser ponderomotive force. The injected

electrons are located at the rear of the bubble and continuously gain energy from the longitudinal field.

The total charge of the injected beam is about 62nC and peak current reaches 9.2kA. Seen from the

directional arrows in the transverse y-z plane, we notice that the transverse fields satisfy BT~-Bϕeϕ and

ET~Erer. Such field distributions indicate a central force on the injected electrons. We show the

trajectories of electrons initially located at a radius of |ri|=6.4μm in Fig. 1(b) and mark their spin

orientations during the injection phase for selected electrons at distinct coordinate angles ϕ=tan-1(z/y)

in the range –π/2 to π/2 (since ϕ+π and ϕ correspond to the same axis with opposite directions, it

suffices to treat the region ϕ∈[-π/2, π/2]). It is seen that due to the cylinder symmetry of the bubble

forces, all electrons are focused to the center, following almost identical trajectories about the central

axis. However, the spin precessions are different for electrons of varying ϕ. In particular, the electron

spin is preserved at ϕ=0 (along the y-axis) during injection while changes dramatically for ϕ=±π/2

(along the z-axis).

Figure 1. (a) A typical LWFA structure and transverse field distributions, with the bubble (green surface), the

laser beam (yellow-purple iso-surfaces for |Ey|=1.5), the injected electrons (black dots), the electron density (the

x–y plane) and the transverse E-field (ET, blue arrows) and B-field (BT, red arrows). The amplitude of

ET=(Ey2+Ez

2)1/2 (x=150μm plane) is projected onto the x=140μm plane. (b) Electron trajectories (solid lines) and

spin orientations (arrows) during injection with ϕ[–π/2, π/2] at injection radius |ri|=6.4μm, with ϕ=0 marked in

blue and the rest in red. The polarization (average spin components in mesh-grids) distributions in the transverse

phase space (py-pz) (c, d and e respectively) along the x, y and z axis at 800fs. The results are collected for

simulation parameters of a=2.5, α=4, n0=1018cm-3 and initial electron spins are set along the z-axis. The cross-

shaped area defined by the phase angle Δϕp in (e) represents the high polarization region.

We note that the trajectories basically remain in a plane defined by the coordinates and the central

axis for each electron. It indicates that the azimuthal angle is the same in both the coordinate space (ϕ)

and the transverse phase space (defined as ϕp=tan-1(Pz/Py)) during the injection. For instance, the

electrons located at ϕ in the range of [-π/2, π/2] would be directed along ϕp~[-π/2, π/2] in the transverse

directions. Therefore, spins should be preserved for electrons moving close to the y-axis (|pz|<<|py|)

and change significantly around the z-axis (|pz|>>|py|). To further prove the statement, we give the

polarization Σsi/N distributions in py-pz space along x, y and z axis (averaging the corresponding spin

components in py-pz mesh-grids). As illustrated in Fig. 1(e) for beam polarization along the +z-axis

(the initial spin orientation), where the polarization purity peaks in the vicinity of ϕp~0 and vanishes

otherwise, exhibiting a saddle-like distribution. The polarization along the y-axis in Fig. 1(d) is also

centrally symmetric but peaks at ϕp~±π/4(|pz|~|py|). The one along the x-axis is symmetric about the

pz=0 axis and averaged out at specific ϕp as shown in Fig. 1(c). Simulations suggest that the peak value

of the polarization along the z-axis is well retained at >80% among the cross region of |ϕp|<π/4, which

allows for spin filtering in the transverse phase space.

To explain the above observations, we consider a quasi-static scenario, where the bubble fields are

cylinder symmetric, i.e. where B~-Bϕeϕ, β~βx, β×E~Erβxeϕ [42-44]. Such field distributions are also

seen in our simulations (Fig.1 (a)). As already stated in previous works [27-29,45], the spin is

preserved during steady acceleration but precesses dramatically during the injection phase. That is

why we focus on the spin dynamics during injection, where typically one has γ<<1/ae [42,46] and the

second term of Eq.(1b) can be neglected. With these simplifications, the precession frequency Ω and

the equation of motion in the transverse direction can be written as:

� = −�

��

��

�+

��

���

��

��eϕ. (2)

���

��(��) = �(�� + �����)�� = ���� . (3)

An obvious effect revealed by Eq. (3) is that the electrons only feel radial forces in transverse directions

∂PT/∂t~∂Pr/∂t. Since electrons injected at an azimuthal angle ϕ carry zero initial azimuthal momentum,

i.e., Pϕ~0, the azimuthal angles are equivalent in the coordinate space and the phase space ϕ~ϕp during

the whole injection process, which agrees with the trajectories in Fig. 1(b). The distributions of injected

electrons with their transverse momentum vectors in Fig. 2(a) further demonstrate that momentum

vectors are almost parallel to the coordinate vector direction, with negligible minor discrepancies. The

difference between the two defined by ϕ–ϕp is counted over all injected electrons, which shows a root-

mean square (RMS) value of only 0.063π in Fig. 2(b).

Known from Eq. (3), the coordinate angle ϕ of an electron is always fixed (there is no lateral force

perpendicular to ��) which guarantees that the precession axis is stationary (along eϕ). The precession

angle satisfies Δθs=<Ω>ΔT where ΔT~4|ri|/c is the injection time and ri is the injection radius [47-49].

The spin vector after injection (starting at t=t0) can then be deduced by s(t0+ΔT)=s//(t0+ΔT)+s ⊥

(t0+ΔT)=s//(t0)+cos(Δθs)s ⊥ (t0)+sin(Δθs)eϕ×s ⊥ (t0), where s//(t0)=[s(t0)·eϕ]eϕ=cosϕeϕ is the component

parallel to Ω and s⊥(t0)= s(t0)-s//(t0)=sinϕer is perpendicular to Ω. One obtains the spin component along

each axis following sx(t0+∆T)=s(t0+∆T)·ex=-sin(∆θs)sinϕ, sy(t0+∆T)=s(t0+∆T)·ey=sin2ϕ[cos(∆θs)–1]/2 and

sz(t0+ ∆ T)=s(t0+ ∆ T) · ez=cos2ϕ+sin2ϕcos( ∆ θs). These results immediately disclose a key feature in

wakefield acceleration: the spin distribution is a function of the coordinate angle ϕ. In the following, we

will show that the dependence holds for the beam polarization.

Figure 2. The number density in the y-z plane (a) and the value of ϕ-ϕp (b) of all injected electrons. The black arrows

in (a) denote transverse momentum-vector directions for randomly selected 1000 electrons. The particle densities in

the ϕp-si space where i=x (c), y (d) and z (e), respectively. The average spin (polarization) of each is shown from the

theoretical analysis (blue-dashed) and the simulations (black solid, calculated from integrating the density along the

si axis). Simulation parameters are the same as those in Fig. 1.

According to Refs. [38,40,50], the average fields in the plasma bubble take the form Er~cBϕ~en(x)r/4ε0.

Taking <βx>~1/2, γ~1, <n(x)>~(α+1)n0/2 and <r>~ri/2 during the injection phase, one obtains <Ω>≈–

5e<Bϕ>/4m≈–5e2(α+1)n0ri/64mε0, thus Δθs~<Ω>~ΔT~–ηri|ri| with η=5e2(α+1)n0/16mc2ε0. For simplicity,

we assume a homogeneous electron distribution within the cylindrical injection volume of |ri|≤rb(xp) [27-

29], where rb(xp) is the bubble radius at the density peak, satisfying rb2(xp)~4σlσr(nb0/αn0)1/2 for PWFA

[42,44] and rb2(xp)~4amc2ε0/αe2n0 for LWFA [51]. Therefore, combining with the fact that ϕ~ϕp, the

polarization at certain ϕp becomes �����/���~�����/�� = ∫ ��|�|/���(��)��

��(��)

���(��), where the three

components (i=x, y, z) are:

�����

���=

�

���(��)

∫ sin(��|�|)sin��|�|����(��)

���(��)= 0 (4a)

�����

���=

������

����(��)

∫ [cos(��|�|) − 1]|�|����(��)

���(��)=

������

�[����(y) − 1] (4b)

�����

���=

�

���(��)

∫ [cos��� + sin���cos(��|�|)]|�|����(��)

���(��)= 1 − sin���[1 − ����(y)] . (4c)

Here ψ is a normalized parameter denoting the largest precession angle under certain parameters, e.g.,

ψ=ηrb2=5e2(α+1)σlσr(nb0n0)1/2/4α1/2mc2ε0 for PWFA and ψ=5a(α+1)/4α for LWFA. The spin distributions

vs. ϕp for the LWFA case are illustrated in Figs. 2(c), 2(d) and 2(e) (ψ=3.91). The theoretical estimation

from Eq. (4) shows good agreement with our simulations. Other cases, like PWFA and LWFA driven by

circularly polarized light, are also well described by Eq. (4).

Equation (4a) is confirmed by the simulation results in Figs. 1(c) and 2(c), where the polarization is

close to 0 along the longitudinal axis (x-axis). At any ϕp the spin values are anti-symmetric around the

sx=0 axis. Although the longitudinal component for each spin can be non-zero due to precession, a pair

of centrally symmetric electrons precesses around opposite axes at the same frequency, resulting in

opposite values of the longitudinal spin component. The polarization along the x-axis naturally

disappears in the statistic average. With regard to the sy distribution, as exhibited in both Figs. 1(d) and

2(d), the polarization peaks at ϕp=±π/4. According to Eq. (4b), both s⊥ and s// contain a factor sin(2ϕp)/2

that contributes to the y component and the maximum absolute value (1–sinc(ψ)) is achieved at π/4.

The highest degree of polarization is preserved in the initial spin orientation. Equation (4c) clearly

shows that the polarization along the z-axis increases when |ϕp| is approaching zero, due to the –sin2ϕp

term contributed by s⊥ and s//. In particular, the polarization reaches 100% at |ϕp|~0 as illustrated in Fig.

2(e). This remarkable feature suggests an efficient method to maximize the beam polarization by

selecting electrons around the phase-space angle ϕp.

We therefore propose the spin filter approach to obtain electron bunches of high polarization purity.

As sketched in Fig. 3(a), the driver beam excites a plasma wakefield in the pre-polarized plasma target,

which accelerates the injected electrons to high energies. After the acceleration, the electron beam can

be filtered by an X-shaped slit. It should be mentioned that the spin distribution in the phase space is

maintained during the long propagation distance after acceleration, because the self-generated field of

the electron beam still follows the form in Eq.(3). The slit is placed perpendicular to x with an opening

angle as, so that electrons with ϕp satisfying -Δϕp/2≤ϕp≤Δϕp/2 (Δϕp=π–as) can pass and all other electrons

are blocked. If we integrate Eqs. (4a) to (4c) we find the total polarization components �����∆��� =

∫ �����∆��/�

�∆��/�/∆�� within the selected angular range

�����∆��� = �����∆��� = 0 (5a)

�����∆��� =�������∆���

�+

�������∆���

�����(y) . (5b)

Such spin filtering only preserves the polarization in the z-direction (the pre-polarized direction). As

exhibited in Fig. 3(b), the simulation results for both LWFA and PWFA are in agreement with Eq. (5).

The polarization decreases with larger values of Δϕp. To be specific, the polarization is 96% for LWFA

and 98% for PWFA at Δϕp=π/10 while the polarization without the spin filter (Δϕp=π) is only 35% for

LWFA and 49% for PWFA.

Figure 3. (a) Sketch of a spin-filter using the X-shaped slit to produce polarized electron beams, where the slit selects

electrons within the region [–Δϕp/2, Δϕp/2]. (b) Beam polarization along the z-axis (Polz) as a function of the selecting

angle Δϕp for LWFA (red solid and cyan square) and PWFA (blue-dotted and cyan circle). The simulation results are

collected with the same parameters as in Fig. 2. (c) Polz as a function of the normalized parameter ψ from simulations

for LWFA (cyan squares) and PWFA (cyan circles) with Δϕp=π/10, π/5, π/2, π. The simulations results are given for

a=0.5~5 for LWFA and σl=0.8~4μm, σr=1.6~3.2μm, nb0=1.5×1019cm-3 for PWFA with α=4 and n0=1018cm-3. The

theoretical predictions are show by lines.

The spin-filter mechanism is universal in a large parameter range. The beam polarization as a

function of the dimensionless parameter ψ is summarized in Fig. 3(c), for different parameter

combinations. While in general smaller selecting angles induce high polarization purity, we see that

the beam polarization is already close to 80% for Δϕp=π/2 (90%, for Δϕp=π/5 and π/10). In other words,

the beam polarization is significantly purified by filtering out only half of the electron flux. Without

spin filtering, the beam flux would be strongly restricted to preserve the polarization. As illustrated in

Fig. 3(c), it ψ<2 is required to maintain a beam polarization above 80% when there is no filtering

(Δϕp=π). Such restrictions disappear in our X-filter strategy. One can find from Fig. 3(c) that for large

value of ψ (representing the beam flux), the beam polarization after spin filtering Δϕp≤π/2 is above

80% throughout the full region.

An interesting phenomenon is that the polarization is not monotone with ψ. As a matter of fact, the z-

component of the spin sz=cos2ϕ+sin2ϕcos(Δθs) oscillates with Δθs, decreasing in the range of [2Kπ,

(2K+1)π] and increasing in the range of [(2K+1)π, (2K+2)π], where K is the integer. The intrinsic of the

spin-filter mechanism is attributed to the field geometry of the wakefield. For transverse initial

polarization, the precession axis is along eϕ, the rotation spin component s⊥ is anisotropic for different

ϕp around the procession axis eϕ. As a consequence, the degree of spin precession depends on ϕp, which

leads to a disparity of polarization in the transverse phase space. However, for longitudinal initial

polarization, the rotation component is |s⊥|=1 for all electrons, which means that the precession degree

is independence of ϕp. We conclude that the spin-filter mechanism is not valid for the longitudinal case.

IV. DISCUSSIONS

The results shown above are based on an ideal situation, where the bubble is in perfect cylindrical

symmetry, the plasma is cold initially, the drive beams are precisely aligned to the filter center and the

filter is regarded as wireless thin. However, in a real experiment, the bubble is usually not so perfectly

symmetric, the plasma gains initially temperature from pre-pulses; the drive beams jitter for high power

laser systems and the filter has a finite size. This section is devoted to the robustness of our method

against the imperfections.

To evaluate imperfect plasma wakefield effects (field asymmetry), we have introduced two

parameters (ρ1, ρ2) in the drive beam profile in our 3D PIC simulations:

�� = �[1 + ���(�, �)]���/��(�)sin�(��/2��)sin(���/��)exp{−[���� + (2 − �1)��)]/��(�)} (6)

where 0<ρ1<2 measures the ellipticity of the transverse intensity profile (perfectly circular for ρ1=1), and

0<ρ2<1 represents the degree of field fluctuation (zero fluctuation at ρ2=0) multiplied by random numbers

uniformly distributed at transverse coordinate δ(y,z) among the range [-1,1]. The latter represents a type

of quite harsh fluctuation that could possibly happen in experiments.

We first investigate the tolerance of parameter ρ1 (ρ2 is set to 0). As shown in Fig. 4(a) and 4(b), for

Δϕp=π/2 (50% filter) the angular dependence of spin distribution is well preserved at ρ1=0.9 but not so

for ρ1=0.8, where the polarization declines in the filter region due to mixing of the spins in asymmetrical

field. From the scanning results depicted in Fig. 2(c), the polarization with Δϕp=π/2 is above 70% for

0.9<ρ1<1.1, indicating a tolerable fluctuation within ±10% for ellipticity. It can be further relaxed to ±20%

(0.8<ρ1<1.2) when choosing a smaller filter Δϕp=π/5. We then show the impacts of field fluctuation by

setting ρ1=1 and varying ρ2 from 0 to 0.25. As illustrated in Fig. 4(d), the polarization declines due to the

uneven distribution of the bubble field. Nevertheless, the polarization still surpasses 70% for ρ2<0.1 when

Δϕp=π/2 and ρ2<0.25 when Δϕp=π/5. These results imply that about 20% fluctuations of laser field

amplitude is tolerable for the filter mechanism.

Figure 4. The polarization (statistics along z axis) distributions in the transverse phase space (py-pz) for (a) ρ1=0.9,

ρ2=0 and (b) ρ1=0.8, ρ2=0 respectively. The polarization along the z-axis as a function of (c) ρ1 and (d) ρ2 for the

selecting angles Δϕp =π/2 and π/5. Other simulation parameters are the same as those in Fig. 1.

As mentioned above, the plasma is not completely cold before arrival of the main pulse. Since the

thermal motion of electrons affect the spin filter process through the mapping from ϕ and ϕp, we carry

out simulations with different initial temperatures (assuming the electrons satisfy the Maxwell-

Boltzmann distribution) under the same parameters as in Fig. 1. As illustrated in Fig. 5(a), the polarization

is well above 70% for temperatures approaching 1keV. This is fulfilled in typical wakefield acceleration

experiments.

In Sec. III, the filter is considered as an ideal plane without thickness. In reality, both transverse

size and longitudinal thickness of our designed filter may affect the selecting polarization. These effects

canbe seen from Monte Carlo simulations, where the target is set 50cm away from the filter. When the

transverse size Lty of the filter increases, the polarization of selecting beam is enhanced while the total

charge declines, as shown in Fig. 5(b). The polarization increases to about 90% when Lty reaches 1cm,

whereas the charge of the selecting beam drops to about 8pC. The filter thickness Ltx has a joint effect

with the beam jitter on the selecting process. In simulations, we artificially introduce a position shift

(along y- and z- axis) between the drive beam axis and the X-filter center. The thickness of the filter is

also varied. The statistics are shown in Fig. 5(c) and (d). The jitterz and jittery represent the beam center

position with respect to (y,z)=(0,0) to the front of the filter. Not surprisingly, one finds the polarization

decline with the augment of both jitter and longitudinal thickness. For thicker filters, the tolerant range

of jitter is smaller. However, the polarization is well beyond 70% if the position jitter<1mm and Ltx<4cm.

These results provide guidance for future experiments.

Figure 5. (a) The polarization along z axis as a function of initial electron temperature. (b) The polarization (blue

squared) and selecting charge (red circled) as a function of the transverse size (along the y-axis) of the filter Lty. The

polarization as a function of longitudinal thickness (along x axis) Ltx and drive beam jitter in z direction (jittery=0)

(c) and in y direction (jitterz=0) (d). The target is set 50cm away from the filter. The selecting angles Δϕp =π/2 while

other simulation parameters are the same as those in Fig. 1.

V. SUMMARY

In conclusion, we investigated the azimuthal dependence of beam polarization in the wakefield and

propose a spin-filter strategy for transversally polarized electron beams. By means of PIC simulations

and theoretical analyses in the scope of a quasi-static model, we find out that the precession degree varies

for certain ϕp (the azimuthal angle in the phase space). In particular, electron spin directions are almost

preserved during acceleration along the ϕp=0 axis (ϕp=π/2 if initially polarized along y axis) since the

rotation components are vanishing. Therefore, applying this effect can maximize the beam polarization

via selection of electrons in certain phase-space regions, at moderate cost of the beam flux. According to

the theoretical model and 3D simulations, about 80% beam polarization can be obtained by filtering out

half of all the electrons, compared to about 35% for the unfiltered beam. The robust of the method is

discussed against various imperfect situations, which will guide future experiments in high power laser

facilities. The filtering mechanism is universal in a large parameter range. The limitations for the driver-

beam parameters and the total beam flux are relieved, which hamper previous schemes. These highly

polarized electron beams are advantageous in applications such as future electron-positron colliders.

ACKNOWLEDGEMENTS

This work is supported by the Strategic Priority Research Program of Chinese Academy of Sciences

(Grant No. XDB 16010000), the National Science Foundation of China (Nos. 11875307, 11935008) and

the Recruitment Program for Young Professionals. MB and AH acknowledge support through the HGF-

ATHENA project.

*jill@siom.ac.cn

#bfshen@mail.shcnc.ac.cn

†ruxinli@mail.siom.ac.cn

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